On the one-sided exit problem for stable processes in random scenery

. We consider the one-sided exit problem for stable Lévy process in random scenery, that is the asymptotic behaviour for T large of the probability where Here W = ( W ( x )) x ∈ R is a two-sided standard real Brownian motion and ( L t ( x )) x ∈ R ,t ≥ 0 the local time of a stable Lévy process with index α ∈ (1 , 2] , independent from the process W . Our result conﬁrms some physicists prediction by Redner and Majumdar.


Introduction
Random processes in random scenery are simple models of processes in disordered media with long-range correlations. These processes have been used in a wide variety of models in physics to study anomalous dispersion in layered random flows [13,5], diffusion with random sources, or spin depolarization in random fields (we refer the reader to Le Doussal's review paper [11] for a discussion of these models). Let us also mention the fact that these processes are functional limits of random walks in random scenery [9,6,7,4,8]. The persistence properties of these models were studied by Redner [15,16] and Majumdar [12]. The interested reader could refer to the recent survey paper [1] for a complete description of already known persistence probabilities and exponents. Supported by physical arguments, numerical simulations and comparison with the Fractional Brownian Motion, Redner and Majumdar conjectured the persistence exponents. In this paper we rigorously prove their conjecture up to logarithmic factors. Before stating our main result, we present the process we are interested in.
Let W = (W (x)) x∈R be a standard two-sided real Brownian motion and Y = (Y t ) t≥0 be a strictly stable Lévy process with index α ∈ (1, 2] such that Y 0 = 0. More precisely, for some positive scale-parameter c, the characteristic function of the random variable Y 1 is given by where γ ∈ [−1, 1]. We will denote by (L t (x)) x∈R,t≥0 a continuous version with compact support of the local time of the process (Y t ) t≥0 . The processes W and Y are defined on the same probability space and are assumed to be independent. We consider the random process in random scenery (∆ t ) t≥0 defined as 2000 Mathematics Subject Classification. 60F05; 60F17; 60G15; 60G18; 60K37. Key words and phrases. Stable process, Random scenery, First passage time, One-sided barrier problem, Onesided exit problem, Survival exponent. This research was supported by the french ANR project MEMEMO2 2010 BLAN 0125. 1 The process ∆ is known to be a continuous δ-self-similar process with stationary increments, with This process can be seen as a mixture of Gaussian processes, but it is neither Gaussian nor Markovian. In this article, we study the asymptotic behaviour of as T → +∞. Our main result is the following one.

Lower Bound
For a certain class of stochastic processes (X t ) t≥0 (to be specified below), Molchan [14] proved that the asymptotic behavior of is related to the quantity We refer to [2] where the relationship between both quantities is clearly explained as well as the heuristics.
Aurzada's proof of the lower bound in the H-index Fractional Brownian Motion (B H (t)) t≥0 case (see [2]) rests on both following arguments: the self-similarity of the FBM and the inequality (valid for a large enough) with C(a) ≤ ca ν , for some c and ν > 0. Our random process ∆ being self-similar, it is enough to prove assertion (2) to derive the lower bound. The increments of the process ∆ being stationary, by self-similarity, we have for every t, s ≥ 0, Conditionally to the process Y , the random variable ∆ 1 is centered Gaussian with variance From the independence of both processes Y and W and from the formula of the even moments of the centered reduced Gaussian law, we can derive the even moments of the random variable ∆ 1 , namely, for any m ∈ N, First of all, from Stirling's formula, for m large enough, we have Moreover, From Corollary 5.6 in [10], there exist positive constants C and ξ s.t. for every λ > 0, So, we have (the constant C may change from line to line but does not depend on m ≥ 1) for some constants C, c > 0. It is now easy to derive (2) namely where C(a) ≤ ca ν with ν := 1 2 (1 + 1 α ).

Upper bound
As in [14] and [2], the main idea of the proof is to bound I(T ) from below by restricting the expectation to a well-chosen set of paths.
Observe that, conditionally to Y , (∆ t ) t is a centered Gaussian process with covariance We will use several times the fact that, due to this fact and to Slepian's lemma, for every 0 ≤ u < v < w and every real numbers a, b, we have and P sup (5) Let p > 0 and β > 0. We define a T := (log T ) p and β T := a For any t > 0, we write |L t | 2 the random variable R L 2 t (x) dx 1/2 . Let us consider the event Lemma 3. For all p > 0 and all β > 0, Proof. First we notice that |L a T | 2 has the same distribution as a 1− 1 2α T |L 1 | 2 and that, by the Cauchy-Schwartz inequality, we have

Hence we have
and so, for T large enough, due to Theorem 4.a in [3], we have (Remark that in the case γ = 1, from (8) in [3], the first probability in (6) is zero and Theorem 4.a [3] can be applied to the Lévy process (−Y t ) t≥0 which is strictly stable with index α and γ = −1). The lemma follows.
Let us define the function By Slepian's lemma (see (4)), we have

Remark that
Conditionally to (Y t ) t , the increments of the process (∆ t ) t being Gaussian and positively correlated, by Slepian's lemma (see (5)), we get Conditionally to (Y t ) t , the random variable ∆ a T is centered Gaussian with variance |L a T | 2 2 and so where F is the distribution function of the Normal distribution N (0, 1). On the event A T , we then have

Moreover,
whereL is the local time of the process (Ỹ t ) t≥0 defined asỸ t := Y a T +t − Y a T for t ≥ 0. Conditionally to (Y t ) t , the processes (∆ t − ∆ a T ) t≥a T and ( L t−a T (x) dW (x)) t≥a T have the same Both probabilities in the right hand side are respectively measurable with respect to the σ-fields σ(Y s , s ≤ a T ) and σ(Y a T +s − Y a T , s ≥ 0) (which are independent one from the other) then we get Let c be the exponent appearing in the lower bound. We choose β > c/(2α) and p such that p(1 − 1 2α − 1 4α 2 ) > 1. Due to Lemma 3 and to the lower bound of F(T ), the first term in the right hand side is larger than C(log a T ) −c a −1/(2α) T for T large enough. The second term is clearly larger than P ∀t ∈ [0, T ], ∆ t ≤ 1 .